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BIT Numerical Mathematics

, Volume 32, Issue 4, pp 620–633 | Cite as

A-stability of Runge-Kutta methods for systems with additive noise

  • Diego Bricio Hernandez
  • Renato Spigler
Part II Numerical Mathematics

Abstract

Numerical stability of both explicit and implicit Runge-Kutta methods for solving ordinary differential equations with an additive noise term is studied. The concept of numerical stability of deterministic schemes is extended to the stochastic case, and a stochastic analogue of Dahlquist'sA-stability is proposed. It is shown that the discretization of the drift term alone controls theA-stability of the whole scheme. The quantitative effect of implicitness uponA-stability is also investigated, and stability regions are given for a family of implicit Runge-Kutta methods with optimal order of convergence.

1980 AMS Subject Classification

65L20 (primary) 60H10,34F05 65L07 93E15 

Keywords and phrases

Numerical stability Runge-Kutta methods implicit methods stochastic differential equations stochastic stability 

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References

  1. 1.
    L. Arnold,Stochastic Differential Equations, Wiley, New York, 1974.MATHGoogle Scholar
  2. 2.
    J. C. Butcher,On the implementation of implicit Runge-Kutta methods, BIT, 16, 237–240 (1976).CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    J. C. Butcher,The Numerical Analysis of Ordinary Differential Equations, Wiley, Chichester, 1987.MATHGoogle Scholar
  4. 4.
    C. C. Chang,Numerical solution of stochastic differential equations with constant diffusion coefficients, Math. Comp. 49, 523–542 (1987).MATHMathSciNetGoogle Scholar
  5. 5.
    J. M. C. Clark, and R. J. Cameron,The maximum rate of convergence of discrete approximations for stochastic differential equations, in B. Grigelionis (ed.),Stochastic differential systems, Lecture Notes in Control and Information Systems, 25, Springer Verlag, Berlin, 1980.Google Scholar
  6. 6.
    G. Dahlquist,A special stability problem for linear multistep methods, BIT, 3, 27–43 (1963).CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    K. Dekker and J. G. Verwer,Stability of Runge-Kutta Methods for Stiff Nonlinear Differential Equations, North Holland, Amsterdam, 1984.MATHGoogle Scholar
  8. 8.
    R. Janssen,Diskretisierung Stochasticher Differentialgleichungen, Preprint Nr. 51, FB Mathematik, Universität Kaiserslautern, 1982.Google Scholar
  9. 9.
    P. E. Kloeden and E. Platen,The Numerical Solution of Stochastic Differential Equations, Springer Verlag, Berlin, 1991.Google Scholar
  10. 10.
    H. Liske and E. Platen,Simulation studies on time discrete diffusion approximations, Math. Comp. Simulation, 29, 253–260 (1987).CrossRefMathSciNetMATHGoogle Scholar
  11. 11.
    G. Maruyama,Continuous Markov processes and stochastic equations, Rend. Circ. Mat. Palermo, 4, 48–90 (1955).CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    E. J. McShane,Stochastic Calculus and Stochastic Models, Academic Press, New York, 1974.MATHGoogle Scholar
  13. 13.
    G. N. Mil'shtein,The Numerical Integration of Stochastic Differential Equations (in Russian), Urals University Press, Sverdlovsk, 1988.Google Scholar
  14. 14.
    E. Pardoux and D. Talay,Discretization and simulation of stochastic differential equations, Acta Appl. Math., 3, 23–47 (1985).CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    W. P. Petersen,Stability and accuracy of simulations for stochastic differential equations, IPS Research Report No. 90-02, ETH-Zentrum, Zürich, January 1990.Google Scholar
  16. 16.
    E. Platen,An approximation method for a class of Ito equations, Litovsk. Matem. Sb. 21 (1981), 121–133.MATHMathSciNetGoogle Scholar
  17. 17.
    W. Rümelin,Numerical treatment of stochastic differential equations, SIAM J. Numer. Anal, 19, 604–613 (1982).CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    J. M. Sancho, M. San Miguel, S. L. Katz and J. D. Gunton,Analytical and numerical studies of computational noise, Phys. Rev. A 26, 1589–1609 (1982).Google Scholar
  19. 19.
    J. M. Varah,On the efficient implementation of implicit Runge-Kutta schemes, Math. Comp., 33, 557–561 (1979).MATHMathSciNetGoogle Scholar

Copyright information

© BIT Foundations 1992

Authors and Affiliations

  • Diego Bricio Hernandez
    • 1
    • 2
  • Renato Spigler
    • 1
    • 2
  1. 1.CIMATGuanajuato, Gto.Mexico
  2. 2.DMMMSA, Università di PadovaPadovaItaly

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